Classes of topological qubits from low-dimensional quantum spin systems

Topological phases of matter is a natural place for encoding robust qubits for quantum computation. In this work we extend the newly introduced class of qubits based on valence-bond solid models with SPT (symmetry-protected topological) order to more general cases. Furthermore, we define and compare various classes of topological qubits encoded in the bulk ground states of topological systems, including SSB (spontaneous symmetry-breaking), TOP (topological), SET (symmetry-enriched topological), SPT, and subsystem SPT classes. We focus on several features for qubits to be robust, including error sets, logical support, code distance and shape of logical gates. In particular, when a global U(1) symmetry is present and preserved, we find a twist operator that extracts the SPT order plays the role of a topological logical operator, which is suitable for global implementation.